A generalization of a theorem of Beardon on analytic contraction mappings
Let \(D\) be a bounded convex domain in \(\mathbb{C}\), and let \(\mathcal F\) be a contracting family of analytic self-mappings of \(D\), i.e., \(|f'(z)|\leq 1\) for \(z\in D\) and \(f\in{\mathcal F}\), and \(\sup_{f\in{\mathcal F}}|f'(z_0)|= 1-c_0\) for some \(z_0\in D\) and some \(c_0>0\). Further, let \(F_n= f_1\circ f_2\circ\cdots\circ f_n\) for \(n\in\mathbb{N}\) and \(f_1,\dots,f_n\in{\mathcal F}\). The author considers the question whether each sequence \((F_n)_{n\in\mathbb{N}}\) converges uniformly to a constant. The key to get such convergence results is to show that the Euclidean diameter of \(F_n(D)\) tends to \(0\). \textit{A. F. Beardon} [Proc. Int. Conf. on Computational Methods and Function Theory, eds. R. M. Ali, St. Rusheweyh and E. B. Saff, 111-128 (1995; per bibl.)] has given the explicit estimate \(\text{diam }F_n(D)= O(1/n^{1/2})\) under the assumption that the curvature of \(\partial D\) is uniformly bounded away from both \(0\) and \(\infty\). For in some sense less strong conditions on \(\partial D\), the author derives an estimate for \(\text{diam }F_n(D)\), which gives, in particular, \(\text{diam }F_n(D)\to 0\), and includes Beardon's result.
scientific article; zbMATH DE number 902532